6. Miscellaneous problems

HermiteHadamard inequality in higher dimensions
The classical HermiteHadamard inequality states that if $f$ is a convex function on $(a, b)$ then $$ \frac{1}{ba}\int_a^b f(x)dx \leq \frac{f(b)+f(a)}{2}. $$ We consider two generalizations to higher dimensions. Let $\Omega \subset \mathbb{R}^n$ be a convex domain and
Case 1: $f\colon \Omega \to [0, \infty)$ convex, or
Case 2: $f\colon \Omega \to [0, \infty)$ subharmonic.
In either case one can ask for smallest constants $c_n, \tilde c_n$ such that \begin{equation} \frac{1}{\Omega}\int_\Omega f(x)dx \leq \frac{c_n}{\partial\Omega}\int_{\partial\Omega}f(x)dx, \quad\mbox{and}\quad \int_\Omega f(x)dx \leq \tilde c_n\Omega^{1/n}\int_{\partial\Omega}f(x)dx. \end{equation} The second inequality follows from the first one by an application of the isoperimetric inequality with $\tilde c_n = c_n/\sqrt{n}$.Problem 6.1.
[S. Steinerberger] Find the optimal constants in the inequalities.
During the workshop improved bounds were found for the sharp constants in all cases [arXiv:1907.06122]. 
Asymptotic partitioning problems
In several problems where one wants to find an in some sense optimal partition of a planar domain $\Omega$ into $N$ subdomains it has been observed that as $N$ becomes large the partition converges to the hexagonal one.Problem 6.2.
[K. Burdzy] Find conditions for problems involving optimal $N$partitions of a planar domain such that one observes in the limit $N\to \infty$ a packing of hexagons. 
The ovals of Benguria and Loss
Let $\Gamma\subset \mathbb{R}^2$ be a closed smooth curve of length $2\pi$ and let $\kappa(s)$ be its curvature as a function of arclength. Define the following selfadjoint onedimensional differential operator \[ H_\Gamma \psi := \psi'' + \kappa^2\psi,\qquad \mathsf{dom}\,H_\Gamma := W^{2,2}(\Gamma), \] in the Hilbert space $L^2(\Gamma)$ and let $\lambda_1(H_\Gamma)$ be its lowest eigenvalue.Conjecture 6.3.
[R. Benguria] The following inequality $\lambda_1(H_\Gamma) \ge 1$ holds. 
FaberKrahn for the buckling problem
Let $\Omega\subset\mathbb{R}^2$ be a bounded domain and consider the lowest eigenvalue of the buckling problem \begin{align} \Delta^2 u+\Lambda_1(\Omega)\Delta u &=0 \quad \mbox{in }\Omega,\\ u&=0 \quad \mbox{on }\partial\Omega,\\ \frac{\partial u}{\partial n}&=0 \quad \mbox{on }\partial\Omega. \end{align}Problem 6.4.
[D. Bucur] Prove that the lowest buckling eigenvalue $\Lambda_1(\Omega)$ is minimized by the disk among all simply connected sets of equal area.
D. Bucur remarks that the problem can be reformulated in terms of the eigenvalue problem \begin{align} \Delta u+ \Lambda_1(\Omega)(uP_\Omega u) &=0 \quad \mbox{in }\Omega,\\ u&=0 \quad \mbox{on }\partial\Omega, \end{align} where $P_\Omega$ is an orthogonal projection from $L^2(\Omega)$ into a subspace of harmonic functions. If the subspace is that of all harmonic functions one retains the eigenvalue of the buckling problem. Partial results exist if one instead projects only onto certain subspaces of harmonic functions (refs?). \ADD
Remark. This buckling load conjecture was raised by G. Polya and G. Szego in their book [MR0043486] on isoperimetric inequalities in mathematical physics.


PoincaréWirtinger extremal domain
Consider the operator $\gamma^{1}\nabla \gamma \nabla$ on the weighted space $L^2(\Omega, \gamma)$ with Neumann boundary conditions, $\Omega\subset \mathbb{R}^n$ and where $\gamma$ is a Gaussian weight. It is known that the first nontrivial eigenvalue of this operator satisfies $\mu_1(\Omega, \gamma)\geq 1$ (this is the PoincaréWirtinger inequality).Problem 6.5.
[D. Krejcirik] Prove that the equality in the PoincaréWirtinger inequality is attained only for infinite strips. 
Nodal structure of the $k$th Neumann eigenfunction for collapsing domains
Consider the nodal domains of the $k$th Neumann eigenfunction on a sequence of domains collapsing to a lower dimensional object.Problem 6.6.
[K. Burdzy] Under what assumptions can one prove that the nodal domains are in a sense ordered?
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Cite this as: AimPL: Shape optimization with surface interactions, available at http://aimpl.org/shapesurface.